Square-Integrable Vector-Valued Functions

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $n\in\mathbb{N}$; $\lVert\cdot\rVert$ is the Euclidean norm on $\mathbb{R}^{n}$.

(i) The function $|f|^{2}$. For a measurable $f:X\to\mathbb{R}^{n}$, $|f|^{2}$ is the function

$$|f|^{2}:X\to\mathbb{R},\qquad x\mapsto\lVert f(x)\rVert^{2}.$$

It is an element of $\mathcal{M}^{+}$ (it is measurable by Sums, Products, Transposes, Dot Products and Norms of Measurable Vector- and Matrix-Valued Maps (iii) and Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i) and has nonnegative values, so it lies in $\mathcal{M}^{+}$ by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi)).

(ii) Square-integrable functions. $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ is the set of measurable functions $f:X\to\mathbb{R}^{n}$ with

$$\int|f|^{2}\,d\mu<\infty,$$

the integral of $|f|^{2}$ as an element of $\mathcal{M}^{+}$. Its elements are the square-integrable functions from $X$ to $\mathbb{R}^{n}$.